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An old problem in combinatorial geometry is to determine when one or more measurable sets in R d admit an equipartition by a collection of k hyperplanes [B. Grünbaum, Partitions of mass-distributions and convex bodies by hyperplanes, Pacific J. Math. 10 (1960) 1257-1261]. A related topological problem is the question of (non)existence of a map f : (S d ) k → S(U ), equivariant with respect to the Weyl group W k = B k := (Z/2) ⊕k S k , where U is a representation of W k and S(U ) ⊂ U the corresponding unit sphere. We develop general methods for computing topological obstructions for the existence of such equivariant maps. Among the new results is the well-known open case of 5 measures and 2 hyperplanes in R 8 [E.A. Ramos, Equipartitions of mass distributions by hyperplanes, Discrete Comput. Geom. 15 (1996) 147-167]. The obstruction in this case is identified as the element 2X ab ∈ H 1 (D 8 ; Z) ∼ = Z/4, where X ab is a generator, which explains why this result cannot be obtained by the parity count formulas of Ramos [loc. cit.] or the methods based on either Stiefel-Whitney classes or ideal valued cohomological index theory [E. Fadell, S. Husseini, An ideal-valued cohomological index theory with applications to Borsuk-Ulam and BourginYang theorems, Ergodic Theory Dynam. Systems 8 * (1988) 73-85].

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It is shown that both the ‘ham sandwich theorem’ and Richard Rado's theorem on general measure (see [6]), which is known to be a measure theoretic equivalent of E. Helly's theorem on convex sets, belong to the same family of results about geometric, extremal properties of measures which are defined on Borel sets in Rn.